Copyright Ó 2007 by the Genetics Society of America
DOI: 10.1534/genetics.106.069450
On the Utility of Linkage Disequilibrium as a Statistic for Identifying Targets
of Positive Selection in Nonequilibrium Populations
Jeffrey D. Jensen,*,1 Kevin R. Thornton,*,2 Carlos D. Bustamante† and Charles F. Aquadro*
*Department of Molecular Biology and Genetics and †Department of Biological Statistics and Computational Biology,
Cornell University, Ithaca, New York 14853
Manuscript received December 12, 2006
Accepted for publication May 19, 2007
ABSTRACT
A critically important challenge in empirical population genetics is distinguishing neutral nonequilibrium
processes from selective forces that produce similar patterns of variation. We here examine the extent to
which linkage disequilibrium (i.e., nonrandom associations between markers) improves this discrimination.
We show that patterns of linkage disequilibrium recently proposed to be unique to hitchhiking models are
replicated under nonequilibrium neutral models. We also demonstrate that jointly considering spatial
patterns of association among variants alongside the site-frequency spectrum is nonetheless of value.
Through a comparison of models of equilibrium neutrality, nonequilibrium neutrality, equilibrium
hitchhiking, nonequilibrium hitchhiking, and recurrent hitchhiking, we evaluate a linkage disequilibrium
(LD) statistic (vmax) that appears to have power to identify regions recently shaped by positive selection.
Most notably, for demographic parameters relevant to non-African populations of Drosophila melanogaster, we
demonstrate that selected loci are distinguishable from neutral loci using this statistic.
P
ATTERNS of DNA sequence polymorphism are
shaped by both a population’s demographic history and natural selection. Uncoupling these two processes is of outstanding importance, as this differentiation
will enable evolutionary biologists to quantify the relative importance of adaptive and nonadaptive factors in
shaping levels of variation in natural populations. A
number of methods have been proposed in recent years
to distinguish demography from selection. For the most
part, research has focused on identifying patterns of
DNA sequence variation that are ‘‘unique’’ to selective
sweeps (Maynard-Smith and Haigh 1974; Hudson
et al. 1987; Kaplan et al. 1989; Tajima 1989; Stephan
et al. 1992; Braverman et al. 1995; Fu 1997; Fay and
Wu 2000; Przeworski 2002; Kim and Nielsen 2004;
Stephan et al. 2006). The most commonly used statistics rely on the site-frequency spectrum (SFS) or observed frequencies of DNA polymorphism in the data.
Predictions are often tested in the form of likelihoodratio tests, comparing a selective sweep model of SFS
variation against (a) a neutral equilibrium model (e.g.,
Kim and Stephan 2002; Kim and Nielsen 2004), (b) a
neutral nonequilibrium model (e.g., Wright et al.
2005), (c) a nonneutral nonequilibrium model (e.g.,
Teshima et al. 2006; Thornton and Jensen 2007), or,
1
Corresponding author: Section of Ecology, Behavior and Evolution,
AP&M 4th Floor Annex, University of California, La Jolla, California
92037. E-mail: jjensen@ucsd.edu
2
Present address: Department of Ecology and Evolutionary Biology,
University of California, Irvine, CA 92697.
Genetics 176: 2371–2379 (August 2007)
with the advent of large-scale genomic data, (d) the
background site-frequency spectrum (e.g., Nielsen et al.
2005). Related approaches have relied on specifically
testing the goodness-of-fit of a given data set to the predictions of a selection model (e.g., Jensen et al. 2005).
The results of this endeavor have been mixed, with
many putatively unique patterns being reproduced by
demographic scenarios.
Since SNP data contain information about linkage
disequilibrium (LD) in addition to site frequencies, it
has been hypothesized that this additional information
could be utilized for hypothesis testing and allow for
greater discriminatory power. Specifically, a number of
theoretical and simulation results have demonstrated
that LD is an important signature of a selective sweep
(e.g., Parsch et al. 2001; Przeworski 2002; Sabeti et al.
2002; Wootton et al. 2002; Kim and Nielsen 2004;
Eberle et al. 2006; Stephan et al. 2006). Therefore, it is
reasonable to think that vital information is being ignored by not considering associations between markers.
The extent to which incorporating LD may improve our
ability to distinguish selection from demography has
been largely unexplored.
Kim and Nielsen (2004) examined the effects of
including LD into the Kim and Stephan (2002) likelihood framework. They describe three patterns of LD
predicted from a genealogical model that are proposed
to be potentially unique to a selective sweep. First, a high
level of LD is expected in regions near, but not immediately adjacent, to the target of selection. Second, a
high level of LD is expected on both sides of the target,
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J. D. Jensen et al.
but should not span the site of selection. Finally, there is
a strong correlation between high-frequency-derived
alleles (as measured by Fay and Wu’s H-statistic) and LD,
such that the probability of observing these alleles is
greater in regions of strong LD. They thus proposed a
new composite-likelihood method designed to incorporate this information. They note, however, that the improvement made by including LD is small, suggesting
that most relevant information is efficiently captured by
considering only the site-frequency spectrum, owing to
the correlation between LD and high-frequency-derived
alleles. Importantly, their result pertains specifically to
the case of distinguishing between a selective and a
neutral equilibrium model.
Stephan et al. (2006) analytically studied a threelocus model of genetic hitchhiking in which one locus is
under positive selection while the other two are neutral
and partially linked. While they further support a number of the conclusions described in Kim and Nielsen
(2004) and further generalize their results, they also note
that when the direction of LD is polarized with respect to
the more common allele at each neutral site, more
positive than negative LD is created after a selective
sweep. They propose that this pattern may indeed be
unique to a selection model, and thus hitchhiking may
have a distinctively patterned LD-reducing effect near the
target of selection. Encouraged by this result, we undertook a simulation study to explore if there were patterns of linkage disequilibrium that are indeed unique to
models of positive selection relative to nonequilibrium
models, which may aid in the discovery of adaptively
important loci.
METHODS
Modeling neutrality: For all neutral simulations we
used Hudson’s (2002) ms program. Specifically, we
simulated data under bottleneck scenarios of varying
intensity as well as under an island model of population
subdivision. We simulated a region of 10-kb-long sequences with a scaled mutation rate of u ¼ 75 and 4Nr ¼
100, where r is the probability per generation of crossing
over for the entire simulated region, values roughly
corresponding to a typical Drosophila melanogaster data
set. The bottleneck model has five parameters: the
population mutation rate (u ¼ 4N0m, where N0 is the
effective size of the ancestral population), the population recombination rate (r ¼ 4N0r), the time at which
the derived population recovered from the bottleneck
(tr), the duration of the bottleneck (d), and the severity
of the bottleneck (f ; 0 , f # 1). In the figures, the time
of the bottleneck (tb) is often referred to—where tb ¼
tr 1 d.
Simulations of population subdivision under an
island model are performed with two subpopulations
and scaled migration rate, M ¼ 4Nm, where m is the
fraction of migrants in each subpopulation in each
generation. The sampling scheme is denoted by n ¼ {n1,
n2}, where n1 and n2 refer to the numbers of chromosomes sampled from the first and second subpopulations, respectively. In this study, we examine equal and
unequal sampling from the subpopulations, for M ¼
0.1, 1, 4, and 10. To distinguish from bottlenecks and
subdivisions, we refer to the model of neutral evolution
under random mating and constant size as the equilibrium neutral model.
Modeling selective sweeps: We model positive selection using coalescent simulations for a region of M
nucleotides. At time t in the past (measured in units of
4N generations), a beneficial allele has fixed in the
population at position X. For all single-sweep simulations, X lies in the interval ½1, M. The simulation consists of a neutral phase, which is the standard coalescent
with recombination (Hudson 1983), and a selective
phase (Braverman et al. 1995). At time t in the past, the
simulation enters the selective phase, which is modeled
as a structured coalescent process (e.g., Kaplan et al.
1988; Braverman et al. 1995), and time is incremented
in small units, dt, until the frequency of the beneficial
allele first reaches x(t) , j, at which point the simulation continues in a neutral phase until the most recent
common ancestor of the sample is reached. Full details
of the single-sweep simulations are found in Thornton
and Jensen (2007).
We also considered a model of selective sweeps occurring in the genome at a rate determined by l, the
expected number of sweeps per recombination unit in
the last 4N generations (Kaplan et al. 1989; Braverman
et al. 1995). Here we allow for selective sweeps both
within the region of M nucleotides as well as at linked
sites. We do this because we simulate a relatively large
neutral region (M ¼ 104), and the probability of a sweep
within that region may not be negligible for large l,
assuming a constant l across the genome. In this model,
the time until the next selective phase is entered is
exponentially distributed with rate 8Nsl=rbp 1 M l,
where rbp is the scaled recombination rate between
adjacent base pairs. The first half of this rate accounts
for sweeps flanking the sequenced region, and the Ml
accounts for sweeps within the region. Given that a
selective phase is entered, the selected site is located
within the M nucleotides with probability M l=ð8Nsl=
rbp 1 M lÞ; otherwise it is located at a linked site up to a
maximum genetic distance of 2a (where a ¼ 2Ns) on
either side of the sampled region (see Kaplan et al. 1989
and Durrett and Schweinsberg 2004 for details).
Briefly, the expected time between successive hitchhiking events is E½tL, the expected length of a hitchhiking event, plus E½tS, the expected time until the next
fixation of a selected allele. For the model considered
here, this equals ðlogj=aÞ 1 1=ð8NsL=rbp 1 M LÞ in
units of 4N generations. For example, for the case of a ¼
5000, l ¼ 105, r ¼ 10, sweeps are occurring on average
every 0:008 time units for the 30-kb region (2a 1 104 1
Selective Sweeps vs. Demography
2373
2a ¼ 30 kb). This extrapolates to approximately one
sweep per 80 generations somewhere in the 120-Mb
euchromatic portion of the D. melanogaster genome.
We estimated LD for two sample sizes (n ¼ 12 and 50)
and 90 parameter combinations generated by considering all combinations of u 2 f10; 75g; r 2 f10; 50; 100g;
a 2 f100; 500; 1000; 2500; 5000g, and l 2 f107 ; 106 ;
105 g. These parameters cover cases where we expect
hitchhiking effects to be minimal (l ¼ 107, a ¼ 100)
to those where the effect should be substantial (l ¼ 105,
a ¼ 5000). For these simulations, we used N ¼ 106.
Statistics: We evaluate the likelihood-ratio test (comparing a neutral equilibrium model and a single-sweep
equilibrium model) proposed by Kim and Nielsen
(2004) under all simulated scenarios. We also examine
the LD statistic that they proposed to more specifically
quantify the extent to which ‘‘sweep-like’’ patterns of LD
are being generated under alternative models. This
statistic, termed v, defined as
1 P
P
l
Sl
2
2
1
r
1
r
i;j2L
i;j2R
ij
ij
2
2
P
v¼
;
ð1=lðS lÞÞ i2L;j2R rij2
divides the S polymorphic sites in the data set into two
groups, one from the first to the lth polymorphic site
from the left and the other from the (l 1 1)th to the last
site (l ¼ 2, . . . , S 2), where L and R represent the left
and the right set of polymorphic sites, and rij2 is the
squared correlation coefficient between the ith and jth
sites. Thus, v increases with increasing LD within each
group and decreasing LD between groups (i.e., the
larger the value of the statistic the more sweep-like the
underlying pattern). For a data set, the value of l that
maximizes v (vmax) is found. Singletons were excluded
prior to calculation. Because the statistic is two tailed,
rejections may be the result of values of vmax that are
either too large or too small relative to the null.
RESULTS
Distinguishing single selective sweep models from
nonequilibrium neutral models: As a starting point, the
Kim and Nielsen (2004) likelihood-ratio test was used
to analyze both neutral nonequilibrium and nonneutral
equilibrium data sets. Parameters for these models were
chosen both for their relevance to natural populations
(particularly for D. melanogaster) and to overlap with the
space investigated in Jensen et al. (2005). When applied
to selection data sets, and consistent with Kim and
Nielsen (2004), we observe that the probability of
rejecting neutrality in favor of selection increases as a
(¼ 2Ns) increases (Figure 1A). An island model with two
subdivided populations was also evaluated, and we
considered a sampling scheme in which all alleles are
sampled from one subpopulation, as well as one in
which the subpopulations are sampled equally. We find
Figure 1.—Results of the Kim and Nielsen (2004) approach incorporating LD when applied to data sets generated
under various models. Results are based on 1000 simulations
for each data point using the following parameter estimates;
4Nr ¼ 100, 4Nm ¼ 75, n ¼ 12, and the length of the region is
10 kb. (A) The time since the sweep is 0.001 in units of 4N
generations, and a ¼ 2Ns. On the y-axis is the probability
of rejecting neutrality in favor of selection (in this case the
proportion of true positives), and on the x-axis is the value
of the selection coefficient a ¼ 2Ns. (B) Values are shown
for two scenarios, one in which all alleles are sampled from
one subpopulation, the other in which alleles are equally sampled. On the y-axis is the probability of rejecting neutrality in
favor of selection (false positives), and on the x-axis is the
value of the migration parameter (4Nm). (C) The time at
which the population recovers back to its prebottleneck size
is 0.0001 in units of 4N generations. Values are shown for
two scenarios, one in which the population is reduced by
90% during the bottleneck and the other in which it is reduced by 99%. On the y-axis is the probability of rejecting
neutrality in favor of selection (false positives), and on the
x-axis is the time of the population crash (tb ¼ tr 1 d) in units
of 4N generations relative to the present.
that the test has a large false positive rate (FPR) (or type
I error) under both scenarios when migration is rare,
and the FPR gradually decreases as 4Nm increases owing
to the deterioration of population structure (Figure 1B).
Finally, we examined neutral stepwise bottleneck
models in which the time of the population crash ranged
from tb ¼ 0.0025 to 0.02 in units of 4N generations
½where tb ¼ time of recovery (tr) 1 duration (d); Figure
1C. False positive rates are observed near 100% for
severe bottlenecks (a 99% reduction in population size),
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J. D. Jensen et al.
Figure 2.—The averages
and standard deviations of
vmax under neutral equilibrium, neutral nonequilibrium, and nonneutral
equilibrium scenarios. Results are based on 1000 simulations for each data point
using the following parameter estimates: 4Nr ¼ 100,
4Nm ¼ 75, the sample size
(n) ¼ 50, and the length
of the region is 10 kb. (A)
A sweep occurred at time
0.002 4N generations ago,
of intensity 500, 1000,
2500, or 5000 in units of
2Ns (where a ¼ 0 is
the equilibrium neutral
model). (B) Two subpopulations are sampled evenly
(n1 ¼ 25; n2 ¼ 25) with rates
of symmetric migration of
0.1, 1, 4, or 10 in units of
4Nm. (C) The population
size is reduced by 90% at
times tb ¼ 0.0025, 0.01,
0.02, or 0.05 4N generations ago and recovered to the prebottleneck size at time 0.0011 4N generations ago. (D) The population
size is reduced by 99% at times tb ¼ 0.0025, 0.01, 0.02, or 0.05 4N generations ago and recovered to the prebottleneck size at time
0.0011 4N generations ago. On the y-axis is vmax.
with slightly lower FPRs for less severe bottlenecks (a
90% reduction in population size). The performance of
this test is thus similar to the CLRT of Kim and Stephan
(2002; Jensen et al. 2005). The parameters examined are
intended to span the non-African bottleneck estimates
recently proposed for D. melanogaster (Thornton and
Andolfatto 2006), although they are relevant for other
recently bottlenecked populations (e.g., humans).
Figure 2 summarizes the averages and standard
deviations of vmax under the equilibrium neutral, nonequilibrium neutral, and equilibrium selection models
examined for n ¼ 50 (n ¼ 12 not shown). There are a
number of notable features. First, under the equilibrium
neutral model (a ¼ 0), vmax-values were observed between 2 and 3 for common (n ¼ 12) and large (n ¼ 50)
sample sizes, with small standard deviations (n ¼ 50
shown in Figure 2A). A number of equilibrium selection
models produced distinctive distributions of vmax. For
large n, vmax is greatest when the selective event was
recent and strong. In contrast, for small sample sizes,
individual observed values of vmax may be reduced relative to the null for large selection coefficients—owing
to the fact that there is very little variation within the
10-kb region following such a severe sweep, an effect that
is exacerbated in small sample sizes.
Second, no model of population structure was identified that regularly produced the pattern of two distinctive stretches of strong LD within, but low LD
between, and the distributions are largely indistinguishable from the neutral equilibrium model (Figure 2B).
Third, modest bottleneck models (90% reduction) returned values of vmax near that observed under neutral
equilibrium conditions even for large sample sizes
(Figure 2C), while severe bottlenecks (99% reduction)
result in a distribution with large values in the tail. For
example, a 99% reduction at time tb ¼ 0.01 4N generations in the past, with a recovery tr ¼ 0.0011 4N generations ago, has an average vmax near 5 (Figure 2D).
Distinguishing nonequilibrium selection models
from nonequilibrium neutral models: The Thornton
and Andolfatto (2006) bottleneck model estimated
for D. melanogaster was singled out for specific analysis.
Results are also presented for sweeps in an equilibrium
population for comparison (Figure 3, a and b). The distribution of the vmax-statistic is largely overlapping between
the neutral and nonneutral scenarios for the bottlenecked population, particularly for n ¼ 12 (Figure 3c).
Thus, these results, taken with those from Figure 2,
strongly suggest that the patterns of linkage disequilibrium proposed to be unique to positive selection are
being replicated under realistic demographic models.
Moreover, these results highlight the relative difficulty
of inferring selection in nonequilibrium vs. equilibrium
populations. The selection distributions of vmax observed under nonequilibrium models are considerably
less distinctive than those under equilibrium models
(e.g., Figure 3c vs. Figure 3a and Figure 3d vs. Figure 3b).
Nonetheless, comparing the neutral and nonneutral
bottleneck models, it is noteworthy that for large sample
sizes (n ¼ 50; Figure 3d), the distributions of vmax are
Selective Sweeps vs. Demography
2375
Figure 3.—Relative densities of vmax under three selection coefficients (a ¼ 0, a ¼ 500, and a ¼ 2500) for (a) a sweep in an
equilibrium population for n ¼ 12, (b) a sweep in an equilibrium population for n ¼ 50, (c) a sweep in a bottlenecked population
for n ¼ 12, and (d) a sweep in the same bottlenecked population for n ¼ 50. 4Nr ¼ 100, 4Nm ¼ 75, t ¼ 0.0041, and the length of the
region is 10 kb. The time of reduction, the severity, and the duration of the bottleneck are taken from the Thornton and
Andolfatto (2006) parameter estimates.
partially distinguishable between neutral and selected
loci in bottlenecked populations. For a ¼ 500, the great
majority of replicates are distinguishable from neutrality, with very large values (vmax . 5) being produced
with high probability. While it is seemingly counterintuitive, larger values of a result in smaller values of vmax,
particularly in the bottlenecked relative to equilibrium
populations. For very large values (a ¼ 2500), the
distribution is still partially distinct from neutrality,
particularly in the direction of values of vmax that are
too small. The twofold diversity-reducing effect of a
bottleneck plus a strong sweep largely eliminates variation within the 10-kb region. Accounting for departures
in both directions, these results indicate that loci that
have experienced recent and strong selection may often
be identifiable in nonequilibrium populations (at least
for the parameter space estimated by Thornton and
Andolfatto 2006), with both small and large values of
vmax being consistent with selection (vmax , 2 and vmax
. 4, respectively). This suggests that the vmax-statistic is
of value when evaluating both African and non-African
sequence data alike.
To better evaluate the utility of the vmax-statistic, we
present receiver operating characteristic (ROC) curves.
In brief, ROC curves plot power as a function of the false
positive rate, where an ideal performance would be a
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J. D. Jensen et al.
Figure 4.—Receiver operating characteristic (ROC) curves for selection in an equilibrium population and a bottlenecked
population. The time of reduction, the severity, and the duration of the bottleneck are taken from the Thornton and
Andolfatto (2006) parameter estimates. 4Nr ¼ 100, 4Nm ¼ 75, t ¼ 0.0041, and the length of the region is 10 kb. We present
results for a ¼ 500, 2500, and 5000 in an equilibrium population for n ¼ 12 (A) and n ¼ 50 (B) and in the bottlenecked population for n ¼ 12 (C) and n ¼ 50 (D). The shaded diagonal line is representative of the situation in which there is an even trade-off
between FPR and power.
curve near the left and the top of the graph (i.e., high
power is achieved with a very low FPR). The diagonal
represents the situation in which there is a linear relationship between power and FPR (e.g., 50% power
corresponds to a 50% FPR). ROC curves are ideal for
these comparisons, as they do not summarize performance merely at a single arbitrarily selected value, but
across all possible values. The ROC curve can be used to
evaluate the gain in power achieved by using a type I
error rate other than the standard 0.05. In particular,
one may prefer to choose a value that balances the
probability of misclassification of either class ½i.e., the
probability of false positives (i.e., type I error) and false
negatives (i.e., power).
Examining ROC curves for our bottleneck with
selection data sets, we observe a number of interesting
features (Figure 4). Once again, results are also presented for sweeps in an equilibrium population for
comparison (n ¼ 12, Figure 4A; n ¼ 50, Figure 4B). For
small sample sizes in a bottlenecked population (n ¼ 12,
Figure 4C), the vmax-statistic has 50% power to detect
strong selection, if an 15% FPR is accepted. Beyond
that point, to increase power, a nearly linear increase in
type I error must be accepted. Notably, for a 5% cutoff,
the test statistic has almost no power. Reiterating a
previous point, because this is a two-tailed test, a number
of these rejections are in the direction of too little LD
relative to the null, particularly for large a. For larger
Selective Sweeps vs. Demography
sample sizes (n ¼ 50, Figure 4D), a different pattern is
observed. A 5% FPR corresponds to 60% power to
detect strong selection. To achieve 80% power the
accepted type I error would approach 30%. For weaker
selection (a ¼ 500), a 5% FPR corresponds to 20%
power. While greater power is achieved with a lower FPR
in equilibrium populations, these results indicate that
vmax is a useful statistic in bottlenecked population as
well, as long as sample sizes are large (n ¼ 50 vs. n ¼ 12).
Distinguishing recurrent-selective-sweep models
from neutrality: As an alternative to the single-sweep
models discussed above, we also consider patterns of LD
produced under recurrent-sweep models. The motivation for considering the recurrent hitchhiking model is
that selective sweeps are a mutation-rate limited process,
and the simulations of a sweep at a particular time do
not account for having to wait for selected mutations to
arise in populations. In particular, we examined parameter combinations relevant for both Drosophila (u ¼ 75,
r ¼ 100) and humans (u ¼ r ¼ 10). Examining both n ¼ 12
and n ¼ 50 across all values, the resulting vmax-values do
not differ significantly from those expected under neutrality (results not shown).
Examining ROC curves better reveals the difficulty of
detecting recurrent selection. Figure 5 plots n ¼ 50, u ¼
75, and r ¼ 10 or 100 (Figure 5, A and B, respectively)
for three rates of sweeps (l ¼ 107, 106, and 105). For
the low-recombination case (Figure 5A), the lowest rate
of sweeps is essentially imperceptible, with a 50% FPR
corresponding to 50% power (solid line in Figure 5A).
For higher rates of sweeps the situation is scarcely better,
with a 5% FPR approaching only 30% power. For highrecombination regions the situation is slightly worse.
Occasionally the performance is poorer than that of the
null model (i.e., the ROC curve is below the diagonal),
owing to the fact that the distribution is contained
completely within that of the null. Basically, for rare
sweeps (solid line in Figure 5B), there is roughly the
same mean, but less variance, in vmax. Thus, these
results indicate that recurrent selection is extremely
difficult to detect using this statistic, as it is for other sitefrequency spectrum-based approaches—particularly for
Drosophila-like recombination parameters. For humanlike recombination parameters, the situation is slightly
better, primarily owing to the fact that lower recombination rates result in stronger patterns of LD across
larger portions of the genome.
DISCUSSION
While bottlenecks have been previously demonstrated to replicate other patterns of the site-frequency
spectrum that are predicted under selection models,
including an excess of high-frequency-derived alleles
(e.g., Przeworski 2002), we here observe that this
includes spatial patterns of variation as well. We propose
that the correlation between high-frequency-derived
2377
Figure 5.—ROC curves for recurrent selective sweeps. We
present results for l ¼ 107, 106, and 105 for n ¼ 50, u ¼ 75,
and r ¼ 10 (A) or r ¼ 100 (B).
alleles and LD observed by Kim and Nielsen (2004)
and Stephan et al. (2006) is generated under neutral
nonequilibrium models as well, with haplotypes that
escape the bottleneck generating stretches of strong
LD. As the bottleneck estimated for non-African populations of D. melanogaster is severe and strongly reduces
diversity, it appears as though the target of selection (l)
in the vmax-calculation is being maximized to regions of
reduced or absent variation, and haplotype blocks that
coalesce during the bottleneck are found in flanking
regions with high probability.
Apart from considering a variety of nonequilibrium
neutral, equilibrium-sweep, and non-equilibrium-sweep
models, we also examine recurrent-sweep models. Although the fixed-t case is of value for quantifying the
performance of these approaches, the recurrent model
is arguably nearer biological reality. While the true rate
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J. D. Jensen et al.
of sweeps in natural populations is unknown, there are
distinct challenges for both the high and low values of l
that we have considered. If the rate of sweeps is high,
then there may be many recent sweeps across the
genome that existing methods will have power to detect.
However, if the rate is this great, then there is an
appreciable probability that sweeps are occurring on
already swept backgrounds. This multiple-sweep effect
will result in very different patterns in the site-frequency
spectrum, particularly with regard to high-frequencyderived alleles (Kim 2006), and thus linkage disequilibrium, owing to the correlation between the two statistics.
If the rate of sweeps is low, then sweeps will be old on
average, and patterns of variability will have recovered
(Przeworski 2002). In other words, a low rate implies
that there will not be many regions of the genome that
have experienced a recent enough sweep to be readily
detectable by existing methods. Thus, the fixed-t singlesweep simulations represent potentially infrequent
evolutionary events. This argument of course relies on
a uniform rate of sweeps over an organism’s recent
evolutionary history. Although this assumption may be
approximately accurate, it is likely violated in a number
of organisms. For example, many domesticated crop
species have experienced very recent and extreme
artificial selection—although the effect of overlapping
sweep patterns will likely be of importance here as well.
Additionally, under this ‘‘domestication’’ scenario, models that consider selection on standing variation will
likely be more relevant than selection on new mutations, a process that results in very different patterns of
variation (Przeworski et al. 2005).
To better evaluate the empirical relevance of the
vmax-statistic, ROC plots were examined for a number of
the most germane scenarios. Most significantly, for the
bottleneck parameters inferred by Thornton and
Andolfatto (2006), the vmax-statistic appears to have
good power to differentiate adaptive loci from neutral
loci in bottlenecked populations. For small sample sizes,
accepting a 15% type I error corresponds to .50%
power to detect selected loci (Figure 4C); and for larger
sample sizes, a 5% type I error corresponds to .60%
power, when selection is strong (Figure 4D). This result
is extremely encouraging, given the difficulty that this
bottleneck parameter space presents for existing and
commonly used test statistics (e.g., Jensen et al. 2005).
For recurrent selective sweeps, the situation appears
less encouraging. Even for strong selection, large
sample sizes, and low rates of recombination, a 5%
FPR corresponds to only 20% power to detect selected
loci (Figure 5A). For other parameter combinations, the
results are essentially near the null (Figure 5B). It is
important to note that the vmax-statistic is designed for
situations in which sequence data span the site of a fixed
beneficial mutation. Thus, under a recurrent selection
model in which sweeps are occurring across a genomic
region that is very large relative to the sampled region, it
may not be surprising that this statistic has low power. As
such, performance will be maximized when the swept
region has been previously localized, as is assumed in
the fixed-t simulations—although it is crucial to account for the ascertainment bias introduced by preselecting regions (Thornton and Jensen 2007). Either
way, the challenges presented for both high and low
rates of recurrent sweeps discussed above remain.
Given the difficulties observed under recurrent selection models when the target of selection has not been
sequenced, we equally anticipate that the vmax-statistic
will have limitations for detecting other types of selection. Specifically, while selection from standing variation
generates patterns that differ strongly from neutrality
(Przeworski et al. 2005), the allele will appear swept
only if the selective pressure began while it was segregating at very low frequency (see Figure 7 of Stephan
et al. 2006). Otherwise the expectation of strong LD
flanking the target, and reduced LD across the target,
described by Kim and Nielsen (2004) and Stephan et al.
(2006), which the vmax-statistic is designed to detect,
will not be created. Additionally, this statistic will likely
be inappropriate to detect partial sweeps. Although this
model may produce strong linkage disequilibrium, owing to the fact that the beneficial allele has undergone at
least part of the rapid increase in frequency, the LD
pattern discussed here is expected to be created only at
the time of fixation. Other LD-based methods have
been proposed that would be more appropriate for
the detection of partial sweeps, such as the extended
haplotype heterozygosity (EHH) approach (e.g., Voight
et al. 2005).
A number of important points need to be mentioned.
First, these results are of particular relevance to derived
populations of species that have experienced a population size reduction associated with colonization. Ancestral populations of these species with stable demographic
histories are less likely to be producing spatial patterns
of variation that replicate sweep predictions, particularly as population structure was not observed to replicate sweep-like patterns of LD. This suggests that
searching for adaptively important loci in these more
stable ancestral populations will likely be fruitful. Apart
from nonequilibrium considerations, however, and given
the recurrent-sweep results, the impact of different rates
of recurrent sweeps (l) needs to be considered when
analyzing empirical data—regardless of whether the
population is ancestral or derived. However, whether
this rate is so great as to obscure individual sweep
patterns, in humans or in flies or in any other natural
population, remains an open question.
Nevertheless, simulations suggest that identifying
adaptively important regions is possible even in bottlenecked populations, despite the fact that we observe
neutral bottleneck models to be capable of producing
patterns of LD previously proposed to be unique to
hitchhiking models. Specifically, loci under strong
Selective Sweeps vs. Demography
selection (a . 500) produce a distribution of vmax that
is only partially overlapping with the neutral case—and
we demonstrate that by accepting a modest type I error
it is possible to achieve significant power. The direction
of the rejection, however, differs along with the intensity of selection, with very strong selection rejecting
because of too little LD relative to the null. These combined results suggest that the vmax-statistic should be
used alongside SFS-based methods when analyzing
polymorphism data and in particular that it appears to
allow for the identification of adaptive loci even in nonequilibrium populations—a challenge that has historically been very difficult to address. If Ne is taken to be on
the order of 1 3 106, selection coefficients on the order
of s ¼ 0.0025 may be identifiable using existing statistics
in ancestral and derived populations alike. Previous
analyses suggest that selection coefficients of this magnitude are not unrealistic for natural populations (e.g.,
Endler 1986). However, the true distribution is unknown, and uncoupling the average strength of sweeps
from the average rate of sweeps remains a formidable
and an important challenge (Wiehe and Stephan 1993;
Kim 2006).
The authors thank Yuseob Kim, Rick Durrett, and two anonymous
reviewers for helpful comment and discussion, as well as the Aquadro
lab. J.D.J. is supported by National Institutes of Health (NIH) grant
GM36431 to C. F. Aquadro and by National Science Foundation grant
DMS-0201037 to R. Durrett, C. F. Aquadro, and R. Nielsen. K.R.T. is
supported by NIH grant GM065509 to A. G. Clark. C.D.B. is supported
by National Science Foundation grant 0516310.
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Communicating editor: D. M. Rand